In the Euclidean space this statement can also be obtained using a
purely geometric argument (see [1]}) and it plays a pivotal
role in the development of the theory of curvature measures for sets of positive
reach in the Euclidean space; see [1]}. However, as mentioned
above, the set \( U \)  might easily be empty even if
\( K = \mathbf {R}^n \sim \Omega \)  and \( \Omega \)  is a convex open subset with
\( \mathcal {C}^{1} \)  boundary, or might reduce to a small tubular neighbourhood
around \( K \)  if \( \Omega \)  has a \( \mathcal {C}^{1,1} \)  boundary. Consequently
Theorem REF  provides no (or very limited) information in these
situations. On the other hand it is well known that the gradient of
\( \mathbf {\delta }^\phi _K \)  is a continuous map on its domain
\( \mathbf {R}^n \sim (K \cup \Sigma ^\phi (K)) \) . Therefore it is a natural
question to ask to identify the largest set on which the gradient of
\( \mathbf {\delta }^\phi _K \)  satisfies a Lipschitz condition. We answer this question
in Theorem REF .
